Optimal. Leaf size=93 \[ -\frac{A \left (a+c x^2\right )^{5/2}}{5 a x^5}-\frac{3 B c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{8 \sqrt{a}}-\frac{3 B c \sqrt{a+c x^2}}{8 x^2}-\frac{B \left (a+c x^2\right )^{3/2}}{4 x^4} \]
[Out]
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Rubi [A] time = 0.152028, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{A \left (a+c x^2\right )^{5/2}}{5 a x^5}-\frac{3 B c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{8 \sqrt{a}}-\frac{3 B c \sqrt{a+c x^2}}{8 x^2}-\frac{B \left (a+c x^2\right )^{3/2}}{4 x^4} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a + c*x^2)^(3/2))/x^6,x]
[Out]
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Rubi in Sympy [A] time = 13.8673, size = 87, normalized size = 0.94 \[ - \frac{A \left (a + c x^{2}\right )^{\frac{5}{2}}}{5 a x^{5}} - \frac{3 B c \sqrt{a + c x^{2}}}{8 x^{2}} - \frac{B \left (a + c x^{2}\right )^{\frac{3}{2}}}{4 x^{4}} - \frac{3 B c^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )}}{8 \sqrt{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+a)**(3/2)/x**6,x)
[Out]
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Mathematica [A] time = 0.28257, size = 104, normalized size = 1.12 \[ -\frac{\frac{\sqrt{a+c x^2} \left (2 a^2 (4 A+5 B x)+a c x^2 (16 A+25 B x)+8 A c^2 x^4\right )}{x^5}+15 \sqrt{a} B c^2 \log \left (\sqrt{a} \sqrt{a+c x^2}+a\right )-15 \sqrt{a} B c^2 \log (x)}{40 a} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a + c*x^2)^(3/2))/x^6,x]
[Out]
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Maple [A] time = 0.017, size = 125, normalized size = 1.3 \[ -{\frac{A}{5\,a{x}^{5}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{B}{4\,a{x}^{4}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{Bc}{8\,{a}^{2}{x}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{B{c}^{2}}{8\,{a}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{3\,B{c}^{2}}{8}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}+{\frac{3\,B{c}^{2}}{8\,a}\sqrt{c{x}^{2}+a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+a)^(3/2)/x^6,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^(3/2)*(B*x + A)/x^6,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.34895, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, B a c^{2} x^{5} \log \left (-\frac{{\left (c x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{c x^{2} + a} a}{x^{2}}\right ) - 2 \,{\left (8 \, A c^{2} x^{4} + 25 \, B a c x^{3} + 16 \, A a c x^{2} + 10 \, B a^{2} x + 8 \, A a^{2}\right )} \sqrt{c x^{2} + a} \sqrt{a}}{80 \, a^{\frac{3}{2}} x^{5}}, -\frac{15 \, B a c^{2} x^{5} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) +{\left (8 \, A c^{2} x^{4} + 25 \, B a c x^{3} + 16 \, A a c x^{2} + 10 \, B a^{2} x + 8 \, A a^{2}\right )} \sqrt{c x^{2} + a} \sqrt{-a}}{40 \, \sqrt{-a} a x^{5}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^(3/2)*(B*x + A)/x^6,x, algorithm="fricas")
[Out]
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Sympy [A] time = 23.3522, size = 199, normalized size = 2.14 \[ - \frac{A a \sqrt{c} \sqrt{\frac{a}{c x^{2}} + 1}}{5 x^{4}} - \frac{2 A c^{\frac{3}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{5 x^{2}} - \frac{A c^{\frac{5}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{5 a} - \frac{B a^{2}}{4 \sqrt{c} x^{5} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{3 B a \sqrt{c}}{8 x^{3} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{B c^{\frac{3}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{2 x} - \frac{B c^{\frac{3}{2}}}{8 x \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{3 B c^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x} \right )}}{8 \sqrt{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+a)**(3/2)/x**6,x)
[Out]
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GIAC/XCAS [A] time = 0.279818, size = 313, normalized size = 3.37 \[ \frac{3 \, B c^{2} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{4 \, \sqrt{-a}} + \frac{25 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{9} B c^{2} + 40 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{8} A c^{\frac{5}{2}} - 10 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{7} B a c^{2} + 80 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{4} A a^{2} c^{\frac{5}{2}} + 10 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} B a^{3} c^{2} - 25 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} B a^{4} c^{2} + 8 \, A a^{4} c^{\frac{5}{2}}}{20 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^(3/2)*(B*x + A)/x^6,x, algorithm="giac")
[Out]